chaotic dynamical system

As Strogatz says in reference [1], “No definition of the term chaos is universally accepted yet, but almost everyone would agree on the three ingredients used in the following working definition”.

Chaos is the aperiodic long-term in a deterministic system that exhibits sensitive dependence on initial conditions.

Aperiodic long-term means that there are trajectories which do not settle down to fixed points, periodic orbits (http://planetmath.org/Orbit), or quasiperiodic as $t\to\infty$ . For the purposes of this definition, a trajectory which approaches a limit of $\infty$ as $t\to\infty$ should be considered to have a fixed point at $\infty$ .

Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast; i.e. (http://planetmath.org/Ie), the system has a positive Liapunov exponent.

Strogatz notes that he favors additional constraints on the aperiodic long-term , but leaves open (http://planetmath.org/OpenQuestion) what form they may take. He suggests two alternatives to fulfill this:

1.

Requiring that there exists an open set of initial conditions having aperiodic trajectories, or
2.

If one picks a random initial condition $x(0)$ then there must be a nonzero chance of the associated trajectory $x(t)$ being aperiodic.

0.1 Further reading

1.

B. Codenotti and Luciano Margara. Chaos in Mathematics, Physics, and Computer Science: Similarities and Dissimilarities. http://pespmc1.vub.ac.be/Einmag_Abstr/BCodenotti.html

0.2 References

1.

Steven H. Strogatz, ”Nonlinear Dynamics and Chaos”. Westview Press, 1994.

Title	chaotic dynamical system
Canonical name	ChaoticDynamicalSystem
Date of creation	2013-03-22 13:05:26
Last modified on	2013-03-22 13:05:26
Owner	bshanks (153)
Last modified by	bshanks (153)
Numerical id	15
Author	bshanks (153)
Entry type	Definition
Classification	msc 37G99
Synonym	chaotic system
Synonym	deterministic chaotic system
Synonym	chaotic behavior
Related topic	DynamicalSystem
Related topic	SystemDefinitions